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Summary form only given as follows. An oscillator of relativistic extended interaction cavity (REICO) is presented and manufactured for the first time. The physics of modulation and RF extraction of REICO is studied in this paper via experiment, theory, and simulation. It is found that the intense relativistic electron beams (IREBs) can be generated intensely current modulation when the IREBs drift through three pillboxes with highly coupling. After bunching in the downstream the modulated IREBs can excited high power microwave in the triaxial cavity. These properties, which have short oscillating time (< 5 ns), compact geometry and high beam-wave conversion efficiency, were encouraged. The mode-competition of modulation was overcome by adjusting the gap width of pillboxes. The output microwave pulse width increasing from 18 ns to 40 ns and the repetition rates increasing from 15 Hz to 100 Hz are in improved after optimizing the structures of the electron beams collection and the triaxial cavity . Using a 900 kV, 16 kA, 45 ns electron beam, 4.1 GW radiated power was extracted in 40 ns FWHM pulses at 2.86 GHz and 100 pps. The power efficiency is 26%, and the energy efficiency is 21%. The average power is 15 kW.